Adjoint functors

In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.
By definition, an adjunction between categories and is a pair of functors
and, for all objects in and in, a bijection between the respective morphism sets
such that this family of bijections is natural in and. For locally small categories, naturality here means that there are natural isomorphisms between the pair of functors and for a fixed in, and also the pair of functors and for a fixed in. For other categories, naturality is defined as a generalisation of this.
The functor is called a left adjoint functor or left adjoint to , while is called a right adjoint functor or right adjoint to . We write.
An adjunction between categories and is somewhat akin to a "weak form" of an equivalence between and, and indeed every equivalence gives an adjunction, though the equivalence itself is not necessarily an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.

Terminology and notation

The terms adjoint and adjunct are both used, and are cognates: one is taken directly from Latin, the other from Latin via French. In the classic text Categories for the Working Mathematician, Mac Lane makes a distinction between the two. Given a family
of hom-set bijections, we call an adjunction or an adjunction between and . If is an arrow in, Mac Lane calls the right adjunct of. The functor is left adjoint to, and is right adjoint to.
In general, the phrases " is a left adjoint" and " has a right adjoint" are equivalent. We call a left adjoint because it is applied to the left argument of, and a right adjoint because it is applied to the right argument of.
If F is left adjoint to G, we also write
The terminology comes from the Hilbert space idea of adjoint operators, with, which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.

Introduction and motivation

Common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right adjoint for a given left adjoint, the fact that left/right adjoint functors respectively preserve colimits/limits, and the general adjoint functor theorems giving conditions under which a given functor is a left/right adjoint.

Solutions to optimization problems

In a sense, an adjoint functor is a way of giving the most efficient solution to some problem via a method that is formulaic. For example, an elementary problem in ring theory is how to turn a rng into a ring. The most efficient way is to adjoin an element '1' to the rng, adjoin all the elements that are necessary for satisfying the ring axioms, and impose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaic in the sense that it works in essentially the same way for any rng.
This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties come in two types: initial properties and terminal properties. Since these are dual notions, it is only necessary to discuss one of them.
The idea of using an initial property is to set up the problem in terms of some auxiliary category E, so that the problem at hand corresponds to finding an initial object of E. This has an advantage that the optimization—the sense that the process finds the most efficient solution—means something rigorous and recognisable, rather like the attainment of a supremum. The category E is also formulaic in this construction, since it is always the category of elements of the functor to which one is constructing an adjoint.
Back to our example: take the given rng R, and make a category E whose objects are rng homomorphisms, with S a ring having a multiplicative identity. The morphisms in E between and are commutative triangles of the form where is a ring map. The existence of a morphism between and implies that S₁ is at least as efficient a solution as S₂ to our problem: S₂ can have more adjoined elements and/or more relations not imposed by axioms than S₁.
Therefore, the assertion that an object is initial in E, that is, that there is a morphism from it to any other element of E, means that the ring R* is a most efficient solution to our problem.
The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor. More explicitly: Let F denote the above process of adjoining an identity to a rng, so F=''R. Let G'' denote the process of "forgetting" whether a ring S has an identity and considering it simply as a rng, so essentially G=''S. Then F'' is the left adjoint functor of G.
Note however that we haven't actually constructed R yet; it is an important and not altogether trivial algebraic fact that such a left adjoint functor actually exists.

Symmetry of optimization problems

It is also possible to start with the functor F, and pose the following question: is there a problem to which F is the most efficient solution?
The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent to the notion that G poses the most difficult problem that F solves.
This gives the intuition behind the fact that adjoint functors occur in pairs: if F is left adjoint to G, then G is right adjoint to F.

Formal definitions

There are various equivalent definitions for adjoint functors:

The definitions via universal morphisms are easy to state, and require minimal verifications when constructing an adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuition involving optimizations.
The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint.
The definition via counit–unit adjunction is convenient for proofs about functors that are known to be adjoint, because they provide formulas that can be directly manipulated.

The equivalency of these definitions is quite useful. Adjoint functors arise everywhere, in all areas of mathematics. Since the structure in any of these definitions gives rise to the structures in the others, switching between them makes implicit use of many details that would otherwise have to be repeated separately in every subject area.

Conventions

The theory of adjoints has the terms left and right at its foundation, and there are many components that live in one of two categories C and D that are under consideration. Therefore it can be helpful to choose letters in alphabetical order according to whether they live in the "lefthand" category C or the "righthand" category D, and also to write them down in this order whenever possible.
In this article for example, the letters X, F, f, ε will consistently denote things that live in the category C, the letters Y, G, g, η will consistently denote things that live in the category D, and whenever possible such things will be referred to in order from left to right. If the arrows for the left adjoint functor F were drawn they would be pointing to the left; if the arrows for the right adjoint functor G were drawn they would be pointing to the right.

Definition via universal morphisms

By definition, a functor
is a left adjoint functor if for each object in there exists a universal morphism
from to. Spelled out, this means that for each object in there exists an object
in and a morphism such that for every object
in and every morphism there exists a unique morphism
with.
The latter equation is expressed by the following commutative diagram:
In this situation, one can show that can be turned into a functor in a unique way such that
for all morphisms in ; is then called a left adjoint to.
Similarly, we may define right-adjoint functors. A functor is a right adjoint functor if for each object in,
there exists a universal morphism from to. Spelled out, this means that for each object in,
there exists an object in and a morphism such that for every object in
and every morphism there exists a unique morphism with.
Again, this can be uniquely turned into a functor such that for a morphism in ; is then called a right adjoint to.
It is true, as the terminology implies, that is left adjoint to if and only if is right adjoint to.
These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem.